In this paper, we show the claims in the original Kipnis-Shamir attack on the HFE cryptosystems and the improved attack by Courtois that the complexity of the attacks is polynomial in terms of the number of variables are invalid. We present computer experiments and a theoretical argument using basic algebraic geometry to explain why it is so. Furthermore we show that even with the help of the powerful new Gröbner basis algorithm like F_4, the Kipnis-Shamir attack still should be exponential but not polynomial. This again is supported by our theoretical argument.

Using the complex coloring method, we present the graphs of the quantum dilogarithm function Gb(z) and visualize its analytic and asymptotic behaviours. In particular we demonstrate the limiting process when the modified Gb(z)→Γ(z) as b→0. We also survey the relations of Gb(z) with different variants of the quantum dilogarithm function.

We study motion of a charged particle confined to a Dirichlet layer of a fixed width placed into a homogeneous magnetic field. If the layer is planar and the field is perpendicular to it, the spectrum consists of infinitely degenerate eigenvalues. We consider translationally invariant geometric perturbations and derive several sufficient conditions under which a magnetic transport is possible, that is, the spectrum, in its entirety or a part of it, becomes absolutely continuous.

We analyze orbifolds with discrete torsion of the ABJM theory by a finite subgroup Γ of SU(2)×SU(2) . Discrete torsion is implemented by twisting the crossed product algebra resulting after orbifolding. It is shown that, in general, the order m of the cocycle we chose to twist the algebra by enters in a non trivial way in the moduli space. To be precise, the M-theory fiber is multiplied by a factor of m in addition to the other effects that were found before in the literature. Therefore we got a Zk|Γ|m action on the fiber. We present a general analysis on how this quotient arises along with a detailed analysis of the cases where Γ is abelian.

We discuss a marginal deformation of the SL(2,R) x SU(2) x U(1)^4 WZW model, which describes string theory on AdS_3 x S^3 x T^4, that corresponds to warping the S^3 factor. This deformation breaks part of the N=(4,4) supersymmetry of the undeformed dual CFT to N=(0,4) supersymmetry. In the spirit of work by Giveon, Kutasov, and Seiberg, we construct the asymptotic spacetime symmetry algebra from worldsheet operators and find a restoration of (4,4) supersymmetry at discrete values of the deformation parameter. We explain this result from various perspectives: the worldsheet, supergravity, and from the singular D1-D5 CFT. The supergravity analysis includes an asymptotic symmetry computation of the level of the affine SU(2) R-symmetry, which arises purely from B-field contributions.